Any company that wants to hire him will have to go through all the hoops of proving there was no suitable Swiss/EU candidate for the position. It's going to be as hard as getting hired from outside Switzerland.
As a citizen of a third country you are here because your employer was able to demonstrate that they were unable to find someone in the labor market to do the job and for no other reason, if you are not meeting that role then there is no reason for you to be here and your employer will want to look for someone to replace you.
Now it may well happen after a couple of L permit renewals you are granted a B permit and later a C permit which would enable you to become sel employed, but that is a long way off.
If he is the main holder, then he definitely can't switch jobs unless they get a completely new approval for him.
If you are here because you got a job and a permit, and he has family reunification, then he's allowed to take up employment (it's subject to approval but as long as he's qualified and the working conditions are appropriate for the position, it's a formality). Because you have L permits, it needs to be as an employee, not as a self-employed contractor. He would however be allowed to start his own GmbH and hire himself as an employee, but that can be an expensive process.
That's not completely correct. Gainful employment of holders of a B permit for family reunification is not limited. So spouses of non-EU B permit holders (or of C permit holders before qualifying for C permit themselves) can register as self employed even though the main permit holder couldn't.